On the triple-error-correcting cyclic codes with zero set {1, 2 i + 1, 2 j + 1}

Abstract

We consider a class of 3-error-correcting cyclic codes of length 2 m - 1 over the two-element field . The generator polynomial of a code of this class has zeroes and , where α is a primitive element of the field . In short, {1, 2 i + 1, 2 j + 1} refers to the zero set of these codes. Kasami in 1971 and Bracken and Helleseth in 2009, showed that cyclic codes with zeroes {1, 2 ℓ + 1, 2 3ℓ + 1} and {1, 2 ℓ + 1, 2 2ℓ + 1} respectively are 3-error correcting, where . We present a sufficient condition so that the zero set {1, 2 ℓ + 1, 2 pℓ + 1}, gives a 3-error-correcting cyclic code. The question for p > 3 is open. In addition, we determine all the 3-error-correcting cyclic codes in the class {1, 2 i + 1, 2 j + 1} for m < 20. We investigate their weight distribution via their duals and observe that they have the same weight distribution as 3-error-correcting BCH codes for m < 14. Further our experiment shows that these codes are not equivalent to the 3-error-correcting BCH code in general. We also study the Schaub algorithm which determines a lower bound of the minimum distance of a cyclic code. We introduce a pruning strategy to improve the Schaub algorithm. Finally we study the cryptographic property of a Boolean function, called spectral immunity which is directly related to the minimum distance of cyclic codes over . We apply the improved Schaub algorithm in order to find a lower bound of the spectral immunity of a Boolean function related to the zero set {1, 2 i + 1, 2 j + 1}. © 2011 Springer-Verlag.